Talk:Polyomino

History etc
Added a sentence of history and the numbers of configurations of various orders. These are the output from a program of my own. Alan Peakall 16:47 Oct 18, 2002 (UTC)

efficient algorithm
I do not know an efficient algorithm. The algorithm that my own program uses is brute force inductive search. It has a heuristic hashing optimization in its implementation. On a 1GHz pentium machine it finds all 63000+ dodecominoes (ie order 12) in under 90 seconds. Alan Peakall 16:31 Nov 7, 2002 (UTC)

2D only
Minor problem: this page says 2D only and is unclear on whether reflections are counted as different. The tetromino page says there are 3D configurations made of cubes; and the pentomino page says mirror-image ones count as the same. -- Tarquin 11:12 Feb 16, 2003 (UTC)

trademark
Major Problem: "Polyomino" is a trademark owned by Professor Solomon Golomb. The mathematically correct word to be used instead is "polysquare". All references to polyominoes in Wikipedia, especially the title of this article, should read "polysquare", not "polyomino". Any use of the word 'Polyomino' should be with the high-'TM' trademark sign only. Authors of this article, please read the Wikipedia conventions on copyright! Editors of Wikipedia, please change all references accordingly. Thank you. Karl Scherer

Cheers, Karl Scherer


 * You say that there's a trademark problem, and that this violates Wikipedia's copyright conventions. Please make up your mind. Thank you. Arvindn 11:46, 16 Mar 2004 (UTC)

Greek or Latin??
Is the -omino suffix Greek or Latin?? 4, 5, 6, 7, 8, 10, and 12 are named with Greek numerical prefixes and 9 and 11 are Latin. Is any set rule being followed here?? 66.245.90.209 00:33, 10 Oct 2004 (UTC)

The set rule is to use Greek prefixes, but 9 and 11 are exceptions in many series of this type, e.g. "nonagon".

article clean up
I have attempted a clean up of the article. The main things I have modified are:
 * structure of the text, headings
 * made more consistent the different definitions of polyominoes: fixed and free, with or without holes
 * added some stats for fixed polyominoes
 * more involved discussion of the group of symmetries between free and fixed polyominoes

I think a few things still need some work: --Bernard Helmstetter 01:47, 24 Dec 2004 (UTC)
 * It would be nice to have more details about the generating functions of the special classes of polyominoes. If it is possible, give expressions of the generating functions, or give the asymptotic growth of the number of polyominoes in those classes
 * The explanation of Conway's method and Jensen's method could be more precise. How are generating functions used?

Terminology - free/fixed?
The terminology on this page seems to be pretty mixed up. By my understanding, free polyominoes can be flipped or rotated, so mirror images and rotations don't count as distinct objects, while fixed polyominoes can not be flipped or rotated. But the table seems to have them the other way round. I'm not sure what the middle column, that say "fixed polyominoes with holes" is actually counting.


 * Yes, I messed it up badly when attempting to clean the article. I think it is corrected now.--Bernard Helmstetter 01:13, 26 Mar 2005 (UTC)

Also, there's no mention of what MathWorld calls one-sided polyominoes, that can be rotated, but not reflected. Since these are the type used in Tetris, it would be nice to enumerate them as well. sjorford &rarr;&bull;&larr; 12:03, 18 Mar 2005 (UTC)

Proposal to delete page "pentominoes"
I think we should remove the page "pentominoes" since there is already a page called "pentomino". Having two articles on the same subject is redundant and not necessary. I moved some of the information in the original "pentominoes" page to the "polyomino" and "pentomino" pages. What are your thoughts on the page deletion? HappyCamper 02:13, 26 Mar 2005 (UTC)


 * ✅. Now pentominoes is a redirect to pentomino. --DavidCary (talk) 20:22, 8 January 2014 (UTC)

Reversed move to polysquare
I've reversed the cut-and-paste move to polysquare, as I disagree with the given reason - that it apparently violates a trademark. A web search for polyomino shows that it is used frequently as a generic noun - not that this is conclusive evidence against a move, but it most definitely needs discussion first. Anyway, there's a move button for doing moves properly, for crying out loud.

I haven't merged anything new back into polyomino, as the articles seemed virtually identical to me. sjorford &rarr;&bull;&larr; 15:42, 31 May 2005 (UTC)

As I recall, Golomb trademarked “pentominoes,” not “polyominoes,” because he  was having plastic sets of pentominoes marketed as toys. See the Wikipedia article on pentominoes. I have seen some writers use “pentaminoes” instead of “pentominoes,” presumably to avoid infringing the trademark. Sicherman 16:09, 9 April 2007 (UTC)

Karl Scherer
Please be aware that Karl Scherer is not a reliable editor. See Votes for deletion/Karlscherer3 for further information. 12:54, 23 Jun 2005 (UTC)

Growth Method issue
The article states the following:
 * This method can be optimized so that it counts each polyomino only once, rather than n times. Starting with the initial square, declare it to be the lower-left square of the polyomino. Simply do not number any square which is on a lower row, or left of the square on the same row. With this improvement, the running time is divided by n, so it only takes about 1 second to enumerate the dodecominoes.

While it seems like a good idea, I think it would fail in the case of any polyomino which did not have any corner squares. The most obvious example would be the cross-piece pentomino. No matter how you rotate it, no square of that will ever occupy the lower-left corner (or any other corner) of the field. Obviously, the greater the value of n, the more polyominos will be omitted using this shortcut. I would hope and assume that anyone who researches this seriously would have taken that drawback into account... Lurlock 04:25, 6 May 2007 (UTC)
 * The "lower-left square" is the leftmost square on the bottom row; in the case of the cross, the (only) square on the bottom square is the lower-left square of the polyomino. This algorithm is that given by Redelmeier (though he has a more efficient method of counting free polyominoes than checking for symmetries after creating each n-omino), and parallel variants have been discussed by other authors; unfortunately the whole article is missing any references, and the figures for time taken are probably original research, as may be the inefficient "Inductive exhaustive search" method (possibly too trivial to have been worth publishing). Joseph Myers 12:11, 6 May 2007 (UTC)

oxford cite
The Oxford English Dictionary cite goes to the wiki article on the dictionary, as opposed to anything showing the eytomology —Preceding unsigned comment added by 194.72.50.160 (talk) 13:28, 12 December 2007 (UTC)

Too Complicated
The concept of polyomino is intuitively simple. This article's lead paragraph gets bogged down in complications. I'm going to try to simplify the introduction. Sicherman (talk) 14:50, 30 June 2010 (UTC)
 * Feel free. The article certainly needs it. --ἀνυπόδητος (talk) 15:51, 30 June 2010 (UTC)
 * Done. Later I may clean up the references if no one beats me to it. Sicherman (talk) 16:33, 30 June 2010 (UTC)

order of a polyomino
In this article, the order refers to the number of cells of a polyomino. This does not match with the "standard" definition of the order as the smallest needed number of copies of the polyomino for tiling a rectangle. — Preceding unsigned comment added by 194.214.127.250 (talk) 06:51, 28 September 2012 (UTC)


 * Weird. Any objections to changing it everywhere to size? —Tamfang (talk) 05:54, 4 October 2012 (UTC)


 * Seven years whiz by ... I'd really like to see some discussion on this. —Tamfang (talk) 20:11, 6 November 2019 (UTC)
 * Making the change now. I'm a mathematician and this is my field of study. This has gotten to the point that web AI (chatgpt/bing) will give out blatantly false information just because a wikipedian mistakenly assumed Order meant Size (no literature supports this idea). 152.117.43.163 (talk) 18:49, 18 May 2023 (UTC)

Where is A_n defined before this symbol is used???
There is a section that deals with the asymptotic growth of the number of distinct polyominoes, which begins:

"''Theoretical arguments and numerical calculations support the estimate

   An  =    c λn/n

where λ = 4.0626 and c = 0.3169."

But as far as I can tell, nowhere previous to this statement is An defined! This despite very careful distinctions made among five categories of polyominoes in the table that gives their counts up through dodecominoes.

Clearly, An is intended to be the count of a certain type of n-ominoes. But which kind?

Even if I missed the definition of An, I think this shows it needs to be defined in the very section giving the asymptotic information, and before An is used in a sentence or a formula.Daqu (talk) 22:13, 28 June 2013 (UTC)
 * I'm sure that An is supposed to be the count of free polyominoes of size n (based on the section it is in). Do you think that the few words I added fix the problem?  DHR (talk) 00:10, 1 July 2013 (UTC)


 * If your presumption of the meaning of An is correct, then your additional wording helps. But it would be far more appropriate -- if the symbol An is going to be used -- to actually define it before using it.  That is, to include a passage like


 * "Let An denote the number of [whatever it means] of size n. Then the following asymptotic formula has been conjectured:"


 * before the asymptotic formula is stated. (The concept of a "conjecture" is different from that of "estimate" (which is irrelevant since we are already talking about asymptotic formulas), and so "conjecture" should be included prior to the formula.


 * Also, what makes you certain that An is the count of *free* polyominoes ? As far as I can tell, the section it is in suggests that it is the count of fixed polyominoes.


 * This also suggests that under the subhead Free polyominoes in that section (even though it would be a simple deduction from the text), it should be mentioned that the same asymptotic formula, with the constant term 1/8 as large as for fixed polyominoes, is conjectured for the counts of distinct free polyominoes.Daqu (talk) 19:43, 2 July 2013 (UTC)

External links modified
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MrOllie, why did you revert the following addition I made in the external links? https://en.wikipedia.org/w/index.php?title=Polyomino&diff=prev&oldid=1138740665 I believe it is a usefull external link, as the site uses compatible terminology with the Wikipedia page while presenting the types and orders of the polyominoes mentioned in the Wikipedia page (free, one-sided, fixed). Pan-wiki-tsik (talk) 16:08, 11 February 2023 (UTC)


 * Wikipedia isn't a link directory, we're not a place to dump links to related applications. MrOllie (talk) 14:18, 12 February 2023 (UTC)
 * Hmmm, after reading the source you provided, it is still not clear to me what WP rule it violates. Let's open a section and see what is the opinion of other wikipedians. Pan-wiki-tsik (talk) Pan-wiki-tsik (talk) 18:10, 13 February 2023 (UTC)

No Tiling Page
There are several natural ways to generalize the polyomino concept that have been explored in the academic literature. These include i) higher dimensional spaces (for example we're working over Z^2 in this article via the duality between the lattice (group) of points and regular unit squares), ii) polyforms (there is an article on these), iii) simply connected, connected, and disconnected tiles (tiles can be thought of as polyominoes or polyforms of some dimension). The most natural operation we do with all of these concepts is produce tilings (i.e. partitions involving fixed types of subsets) of certain types of mathematical spaces (e.g. Z^n or R^n) using certain types of tiles (e.g. subsets of Z^n or convex polygons). However any key organizing principle involving this tiling concept seems to be presently lacking. I propose that we create a general "Tile" or "Tiling" (mathematics) page that brings together all of these concepts. Child topics being linked to by this page could include things like Tiling in the plane and also Polyominoes.

Also there is some literature involving disconnected polyominoes (e.g. in Z^n) absent from Wikipedia so far (e.g. 'Tiling with arbitrary tiles' Gruslys et al. 2015)

- Mathguy9109 (talk) 23:21, 16 June 2018 (UTC)


 * There is an existing article, Tessellation, on the subject of tilings, which is, in fact, the top link on the disambiguation page Tiling. Furthermore, Polyomino already links to the tessellation article.  See the second sentence of "Uses of polyominoes". I would add that many aspects of the study of polyominoes have nothing to do with their ability, or lack thereof, to tile the plane.  So while there is certainly some overlap between the topic of polyominoes and the topic of tilings, neither belongs under the umbrella of the other. Will Orrick (talk) 00:17, 17 June 2018 (UTC)


 * I suppose that page will suffice as a hub. Under "Uses of Polyominoes" all of the cases mentioned are related to tiling. Maybe you could add some of these aspects independent of tilings? Anyway as it stands right now there is only a very tiny blurb about polyominoes on the Tessellation page. I suppose I could write about these results that I mentioned before either on the Polyomino page or alternatively use them to expand the Tessellation page. Which of these locations do you think you'd prefer? - Mathguy9109 (talk) 02:53, 17 June 2018 (UTC)


 * I should have been a bit clearer: all I really meant is that a large fraction of the existing article is concerned with enumeration of polyominoes according to number of squares, which is a separate issue from ability to tile a region or the plane, and which is important in the physics and chemistry applications. I don't have any other material to add at present, but I did move the tiling material to its own section of the article and moved the remaining small amount of material under "Uses of polyominoes" under the new heading "Polyominoes in puzzles and games".  If you want to add some material about tilings with generalizations of polyominoes, my opinion is that that material fits much better in the polyomino article rather than the tessellation article. Will Orrick (talk) 05:55, 18 June 2018 (UTC)


 * Another reason the Tessellation page doesn't particularly appeal to me is because it's an amalgam of strictly mathematical topics, but also has separate sections on art, manufacturing, and nature; most of the time we just use something like Tiling (mathematics) to disambiguate instead of lumping together a bunch of topics which fall under these different parent categories. I would further draw a distinction between Discrete Tiling and Tiling by Polygons, as the literature/techniques here are very different. Afterward I would include both the Polyomino/Polyform pages under this Discrete Tiling category. - Mathguy9109 (talk) 11:35, 20 June 2018 (UTC)


 * Although I have an interest in and have done a small amount of work on tessellations, I don't feel that I am enough of an expert to have an opinion on your suggestions. If you want to do major surgery on Tessellations, it would be a good idea to solicit opinions on the talk page of that article first.  Whatever the flaws of the Tessellations article, I don't think it would be a good idea to create a parallel set of Tilings articles that has substantial overlap with the Tessellations article, although if you can establish a consensus among the editors of that article, it might make sense spin off the strictly mathematical material into a separate article, leaving only a synopsis and link in the main article.


 * Does Wikipedia have a notion of categories and subcategories? I'm not sure that it does, but if so, I still don't feel that it makes sense for Polyomino to be a subcategory of Tessellation.  To my mind, that would be like making Polygon or Polyhedron a subcategory of Tessellation (because they can be used in tessellations).  There really is a substantial literature on enumeration of polyominoes as a function of size and asymptotic growth questions, with application to statistical physics, that has little to do with tiling problems.  Since quite a bit of attention has been paid to tiling questions, that will certainly affect how the article is written, but it shouldn't affect how it logically relates to other topics.  Will Orrick (talk) 14:39, 20 June 2018 (UTC)

Misformatted Web References
The Resta and Zucca references have been reformatted to use the web page reference template. On the whole, this is good, because it gives dates for archival and verification. But the references render with extra quotation marks. Can this be corrected? Sicherman (talk) 11:09, 18 February 2022 (UTC)

New external link suggestion
I suggest the following site to be added in the externa links: Polyomino Maker The suggested link displays on the browser window all the polyominoes of the main page: Enumeration of polyominoes, namely free (total), one-sided and fixed with the correct enumeration in each category. I think it also gives the ability to download the polyominoes on the user's device, but I haven't checked this option out yet.

This external link is consistent with the Wikipedia rules found at Wikipedia:External_links Specifically, it seems consistent with the following rules mention in the above page: What to link
 * Is the site content proper in the context of the article (useful, tasteful, informative, factual, etc.)?

What can normally be linked 3. Sites that contain neutral and accurate material that is relevant to an encyclopedic understanding of the subject and cannot be integrated into the Wikipedia article due to copyright issues,[4] amount of detail (such as professional athlete statistics, movie or television credits, interview transcripts, or online textbooks), or other reasons.

Fellow wikipedians please share your opinion. Pan-wiki-tsik (talk) Pan-wiki-tsik (talk) 18:14, 13 February 2023 (UTC)


 * As I wrote above, Wikipedia isn't a link directory, we're not a place to dump links to related applications. Software tool listings are off topic for an encyclopedia. MrOllie (talk) 18:28, 13 February 2023 (UTC)
 * I agree with MrOllie. External links can be useful when there is some unique resource related to the article's subject—for example the subject's personal web page when the article is a biography, or an organization's official website for an article about that organization. For an article about a scientific area of study an appropriate external link might be a standard database used by practitioners in that area.
 * On the other hand, when there is a larger number of smaller, more specialized resources, and no recognized standard, what tends to happen to articles that aren't carefully watched is that they accumulate lists of random, competing links. I don't believe this is terribly useful for readers, who can get more comprehensive results using a search engine, and I don't think it's a very good look for Wikipedia. Will Orrick (talk) 19:36, 13 February 2023 (UTC)